Evaluation of anomalous scattering factors, <IMG ALIGN=BOTTOM SRC="_6400_tex2html_wrap9102.gif"> and <IMG ALIGN=BOTTOM SRC="_6400_tex2html_wrap7585.gif">.



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Evaluation of anomalous scattering factors, and .

 

The method used for the determination of was that suggested by Hoyt et al [58]. The calculations are based on Eq. gif. This expression contains a singularity at and an integral running from which is numerically impossible to evaluate. However Hoyt et al showed that with certain approximations the singularity may be conveniently dealt with and that by suitable choice of upper and lower integration limits a numerical solution can be found.

Modifications were necessary to the method of Hoyt et al due to the fact that the data in the present work was based on fluorescence measurements whereas transmission EXAFS data were used for the original work. Secondly the criteria established for the choice of the upper and lower energy integration limits were relevant to K-edge data and had to be re-established for data around the edge.

Fluorescence data measured around the absorption edge of interest from the crystal sample is first spline fitted and normalised such that the resulting values are unity far above the edge and zero far below the edge with the edge structure seen as fluctuations about unity. Several possibilities exist for converting such experimental data into an curve. The data may be fit directly to theoretical data for or it may first be fit to atomic cross section data and in turn converted to using the Optical theorem. However difficulties in measuring absorption thicknesses tend to introduce systematic error into atomic cross section data [111]. For this reason the first method was used for fitting the experimental data in the region of the absorption edge but since the available data did not extend to very high or very low energies tables of atomic absorption data were used for the extrapolation to these energies.

In the vicinity of the absorption edge the normalised experimental fluorescence data were converted into data by multiplying it into theoretical data of Cromer & Libermann [25] calculated using CROSSEC. Since the experimental data only extended out to about a hundred electron-Volts above and below the edge, a straight line fit to the theoretical data was performed for the above and below edge regions to produce two linear representations of , and as a function of energy. The experimental curve was calculated by

Far above and far below the absorption edge the normalised data were converted into the atomic cross-section, , in the following way. Two forms of a - type equation for the mass absorption coefficient above and below the absorption edge were determined by inserting the coefficients , given in the McMasters tables [85], into the following Eq. [59] [85]

 

where is the atomic cross-section for energies above the absorption edge (corresponding to , , , ). The values of were summed at the appropriate energies to yield the total cross-section . Experimental values for were then derived from cross-sections using the Optical theorem (see Sec. gif, Eq. gif)

The Kramers-Kronig relationship, Eq. gif, may be written in terms of energy as

 

The presence of the singularity at requires that the equation be manipulated to allow integration numerically. Hoyt et al. split the integral into three parts and deal with the singularity using a Taylor series expansion. The resulting numerically calculable solution is,

 

The integration was carried out using Simpson's rule. The last term of Eq. gif was summed only up to and including the term since higher terms were of the order of of the overall magnitude of the summation term. An additional correction term equal to , where is the total energy of the atom in question, has been determined by Cromer & Libermann [25] and is added to the resulting value. Values of for all atoms between and are tabulated in Table. II of [25].

The resulting program was initially tested with K-edge EXAFS data collected on an copper foil. The data were collected on the EMBL EXAFS beam-line in HASYLAB, DESY with an Si(111) monochromator in the absence of any focussing mirror. The energy resolution of the beam line at the time was measured to be . The data were placed on an absolute energy scale [93]. To carry out a comparison with the copper K-edge data published by Hoyt et al the same integration limits were used, i.e. an upper integration limit of and a lower limit equal to .

The experimental curve of Hoyt et al and that obtained from the experimental data of Pettifer were found to be very similar in shape with only very minor differences in the fine structure. However, it was observed that the data of Hoyt et al were on an arbitrary energy scale shifted with respect to calibrated data by . This still allowed an idea to be obtained about the relative magnitudes of the resultant data by applying an approximate energy correction to the data of Hoyt et al.

  
Figure: Comparison of the results of the Kramers-Kronig transformation performed on data taken around the K-edge of copper. The solid line represents the results of the present transformation program using data collected at the EMBL EXAFS beam-line at an energy resolution of and placed on an absolute energy scale. The points are taken from the data published by Hoyt et al. Ignoring the mismatch in the energy axis there is discrepancy of about in the value.

The results from both studies are shown in Fig. gif. The solid line in the result of the present calculation and the set of data points shown are taken from the results of Hoyt et al. The mismatch in energy of both sets data is clearly visible. When the data of Hoyt et al were shifted along the energy axis and along the axis the best fit was obtained at values corresponding to a energy shift of and . The agreement between the two sets of results is to better than if the discrepancy along the energy axis is overlooked.

Having established that the results produced by the present program were in sufficient agreement with those of the Hoyt et al, it was then necessary to re-establish the integration criteria for use with edge data. A similar procedure as that used in the previous work was followed in that several upper and lower limits of integration were used and the point of convergence for the calculated data was sought. The data used for this purpose were Pt absorption edge data collected on the EMBL EXAFS beam-line from a sample of ).

  
Figure: Resulting curves when the lower integration limit is held at and upper limits of (top curve), (middle curve), , and edge energy are used. The lower curve is produced by the latter three of these demonstrating that convergence can be achieved by using an upper integration limit of edge energy.

Figures gif and gif show respectively the curves calculated for various values of the upper and lower integration limits. Convergence of the calculated values was achieved when using an upper integration limit of and a lower limit of .

  
Figure: Results of the K-K transformation when the upper integration limit is kept at edge energy and lower limits of , , , , and edge energy (from bottom to top). The upper most curve arises from limits of and thus demonstrating that a value of is sufficient as a lower integration limit.

These integration limits were then used together on the cis-platin data and the resulting curve was compared with the curve obtained by using the theoretical values for obtained from the theory of Cromer & Libermann [25] implemented in the program CROSSEC. The two curves are shown in Fig. gif. The deviation of the experimental curve from theory is a result of the structure present in the XANES region of the absorption spectrum.

  
Figure: Plot comparing values from the result of the Kramers-Kronig transformation performed on experimental absorption data taken at the edge of platinum in cis-platin (solid line) with those value calculated using the theory of Cromer & Libermann (dashed line).



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Next: Experimental Apparatus. Up: Theory and methods Previous: Model refinement.



Gwyndaf Evans
Fri Oct 7 15:42:16 MET 1994